We study via theory and dynamical simulation the evolving structure, particle dynamics, and time-dependent rheological properties of an aging colloidal gel, with a focus on the micro-mechanics that drive coarsening and age-related changes in linear-response behavior. When colloids in suspension attract one another, the attractions can lead to phase separation into particle-rich and particle-poor regions separated by a single interface. But this transition is sometimes interrupted before full separation occurs. With certain particle concentrations and interparticle potentials, the attractions that promote phase separation also inhibit it, frustrating the separation and “freezing in” a nonequilibrium particle configuration, resulting in a space-spanning gel. With attractions on the order of a few kT, gelation can produce nonfractal bicontinuous morphologies. In such “reversible” gels, thermal fluctuations are strong enough to rupture bonds and reform new ones, allowing restructuring of the gel over time. But, because particle diffusion is dramatically slowed by interparticle attractions, the march toward equilibrium is frustrated. Prior studies of colloidal gels have examined evolution of length scales and dynamics such as decorrelation times or heterogeneity. Left open were additional questions such as how the particle-rich regions are structured (liquidlike, glassy, and crystalline), how restructuring takes place (via bulk diffusion, surface migration, and coalescence of large structures), and the impact of the evolution on rheology. In this study, we conduct dynamic simulations to elucidate the post-gelation evolution of a system of 750 000 Brownian spheres interacting via a hard-sphere repulsion and short-range attractions of order kT, as would be generated by a polymer depletant, for example. We find that the network strands comprise a glassy, immobile interior near random-close packing, enclosed by a liquidlike surface along which the diffusive migration of particles drives coarsening. We show that coarsening is a three-step process of cage forming, cage hopping, and cage arrest, where particles migrate to ever-deeper energy wells via “Smoluchowski's ratchet.” Both elastic and viscous high-frequency moduli are found to scale with the square-root of the frequency, similar to the perfectly viscoelastic behavior of nonhydrodynamically interacting, purely repulsive dispersions. But here, the behavior is elastic over all frequencies, with a quantitative offset between elastic and viscous moduli which owes its origin to the hindrance of diffusion by particle attractions. Propagation of this elasticity via the network gives rise to age-stiffening as the gel coarsens. This simple phenomenological model suggests a rescaling of the moduli on network length scale which, when carried out, collapses each modulus for all gel ages onto a single universal curve. A theoretical model inspired by the Rouse model is advanced and, from it, we have obtained an analytical expression that captures the effects of (finite) structural aging on rheology: The moduli are linear in the network size, suggesting that linear mechanical response can be determined at any age by measurement of dominant network length scale—or vice versa.
The authors acknowledge the contribution of undergraduate researcher Christopher Canova for his work in carrying out rheology simulations. The authors wish to acknowledge generous support of the National Science Foundation, without whose access to the computational resources of XSEDE of this study would not have been possible. The authors also thank the Princeton Institute for Computational Science and Engineering for its generous support. This work was supported in part by National Science Foundation Grant No. CBET-0754078, by the Cornell University Atkinson Center for a Sustainable Future under Grant No. U268703-706, and by the Cornell University Engineering Learning Initiatives Program. R.N.Z. wishes to thank George Porter, Engineering Librarian at the California Institute of Technology, for invaluable help locating original and translated articles of Einstein, Perrin, and von Smoluchowski.
I. INTRODUCTION II. MODEL SYSTEM III. DYNAMIC SIMULATION METHOD IV. RESULTS A. Structure B. Dynamics C. Smoluchowski's ratchet 1. Role of interparticle attraction D. Rheology 1. Linear-response moduli 2. Experimental measurement V. DISCUSSION AND CONCLUSIONS